Two-sample Statistics Based on Anisotropic Kernels
Two-sample Statistics Based on Anisotropic Kernels
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DOI：
10.1093/imaiai/iaz018
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发表时间：
2017-09
期刊：
影响因子：
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通讯作者：
Xiuyuan Cheng;A. Cloninger;R. Coifman
中科院分区：
文献类型：
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作者：
Xiuyuan Cheng;A. Cloninger;R. Coifman
本文介绍了一种新的基于核的最大平均偏差统计量，用于度量给定有限多个多元样本的两个分布之间的距离。当分布是局部低维分布时，通过结合局部协方差矩阵和构造各向异性核，所提出的检验方法可以更有效地区分某些备选方案。核矩阵是不对称的；它计算[公式：请参阅文本]数据点和一组[公式：请参阅文本]参考点之间的亲和力，其中[公式：请参阅文本]可以大大小于[公式：请参阅文本]。虽然所提出的统计量可以被看作是一类特殊的再生核Hilbert空间MMD，但在对核的较温和的假设下，只要[公式：见正文]，并且得到了检验功率的有限样本下界，则证明了检验的相合性。应用于流式细胞术和弥散磁共振数据集，这激励了所提出的方法来比较分布。
The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between [Formula: see text] data points and a set of [Formula: see text] reference points, where [Formula: see text] can be drastically smaller than [Formula: see text]. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as [Formula: see text], and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.